lsatcvat - Mathbox for Norm Megill

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Theorem lsatcvat 40087

Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 32981 analog.) (Contributed by NM, 10-Jan-2015.)

**Hypotheses**
Ref	Expression
lsatcvat.o	⊢ 0 = (0_g‘𝑊)
lsatcvat.s	⊢ 𝑆 = (LSubSp‘𝑊)
lsatcvat.p	⊢ ⊕ = (LSSum‘𝑊)
lsatcvat.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lsatcvat.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lsatcvat.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lsatcvat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
lsatcvat.r	⊢ (𝜑 → 𝑅 ∈ 𝐴)
lsatcvat.n	⊢ (𝜑 → 𝑈 ≠ { 0 })
lsatcvat.l	⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))

**Assertion**
Ref	Expression
lsatcvat	⊢ (𝜑 → 𝑈 ∈ 𝐴)

**Proof of Theorem lsatcvat**
Step	Hyp	Ref	Expression
1		lsatcvat.o	. . 3 ⊢ 0 = (0_g‘𝑊)
2		lsatcvat.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lsatcvat.p	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
4		lsatcvat.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑊)
5		lsatcvat.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
6	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec)
7		lsatcvat.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
8	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
9		lsatcvat.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
10	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
11		lsatcvat.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	11	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
13		lsatcvat.n	. . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 })
14	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 })
15		lsatcvat.l	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
16	15	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
17		simpr 490	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈)
18	1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17	lsatcvatlem 40086	. 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
19	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec)
20	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
21	11	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
22	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
23	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 })
24		lveclmod 21374	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
25	5, 24	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
26		lmodabl 21177	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
27	25, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel)
28	2	lsssssubg 21226	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
29	25, 28	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
30	2, 4, 25, 9	lsatlssel 40034	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
31	29, 30	sseldd 3932	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
32	2, 4, 25, 11	lsatlssel 40034	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
33	29, 32	sseldd 3932	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
34	3	lsmcom 20065	. . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
35	27, 31, 33, 34	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
36	35	psseq2d 4044	. . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄)))
37	15, 36	mpbid 235	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
38	37	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
39		simpr 490	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈)
40	1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39	lsatcvatlem 40086	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
41	29, 7	sseldd 3932	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
42	3	lsmlub 19871	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
43	31, 33, 41, 42	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
44		ssnpss 4055	. . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))
45	43, 44	biimtrdi 256	. . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)))
46	45	con2d 135	. . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈)))
47		ianor 997	. . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
48	46, 47	imbitrdi 254	. . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)))
49	15, 48	mpd 16	. 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
50	18, 40, 49	mpjaodan 973	1 ⊢ (𝜑 → 𝑈 ∈ 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ⊆ wss 3899 ⊊ wpss 3900 {csn 4584 ‘cfv 6537 (class class class)co 7418 0_gc0g 17603 SubGrpcsubg 19323 LSSumclsm 19841 Abelcabl 19988 LModclmod 21128 LSubSpclss 21199 LVecclvec 21370 LSAtomsclsa 40011

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-tpos 8236 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-0g 17605 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-grp 19140 df-minusg 19141 df-sbg 19142 df-subg 19326 df-cntz 19524 df-oppg 19553 df-lsm 19843 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-drng 20975 df-lmod 21130 df-lss 21200 df-lsp 21240 df-lvec 21371 df-lsatoms 40013 df-lcv 40056

This theorem is used by: lsatcvat2 40088

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